criteria for existence of antidervatives
Let be a normed space, a Banach space, a connected open set, a continuous function, where is the space of continuous linear operators. In this article a path is a curve that has bounded variation. The following theorems give necessary and sufficient conditions for to have an antiderivatives.
The next theorem states criteria for the existence of local antiderivatives.
The following conditions are equivalent:
has an antiderivative locally,
for , homotopic closed paths in ,
if is a triangular path such that its convex hull is in , then .
where is the space of bilinear operators from to . If is an antiderivative of , then and by Clairaut’s theorem the second derivative is symmetric. The following theorems assert that the reverse is also true.
If is differentiable, then it has an antiderivative locally if and only if is symmetric for all .
Combining these three theorems immediately gives the following.
If is simply connected and is differentiable, then it has an antiderivative on if and only if is symmetric for all .
|Title||criteria for existence of antidervatives|
|Date of creation||2013-03-22 19:14:00|
|Last modified on||2013-03-22 19:14:00|
|Last modified by||scineram (4030)|